Quite simply, cutting a customer off is about the worst (and rudest) thing customer support can do, even if it's automated. Instant anger and frustration leading to the potential for them to take their business elsewhere.
Makes a mockery of the whole "your call is valuable to us" business
(though that's always been a joke in my opinion).
I'd even be happy to be routed to India than be cut off. At least I get to speak to someone even if neither of us can understand one another.